Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > iordsmo | Unicode version |
Description: The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.) |
Ref | Expression |
---|---|
iordsmo.1 |
Ref | Expression |
---|---|
iordsmo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresi 5235 | . . 3 | |
2 | rnresi 4891 | . . . 4 | |
3 | iordsmo.1 | . . . . 5 | |
4 | ordsson 4403 | . . . . 5 | |
5 | 3, 4 | ax-mp 5 | . . . 4 |
6 | 2, 5 | eqsstri 3124 | . . 3 |
7 | df-f 5122 | . . 3 | |
8 | 1, 6, 7 | mpbir2an 926 | . 2 |
9 | fvresi 5606 | . . . . 5 | |
10 | 9 | adantr 274 | . . . 4 |
11 | fvresi 5606 | . . . . 5 | |
12 | 11 | adantl 275 | . . . 4 |
13 | 10, 12 | eleq12d 2208 | . . 3 |
14 | 13 | biimprd 157 | . 2 |
15 | dmresi 4869 | . 2 | |
16 | 8, 3, 14, 15 | issmo 6178 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wcel 1480 wss 3066 cid 4205 word 4279 con0 4280 crn 4535 cres 4536 wfn 5113 wf 5114 cfv 5118 wsmo 6175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-smo 6176 |
This theorem is referenced by: smo0 6188 |
Copyright terms: Public domain | W3C validator |